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A058711 Triangle T(n,k) giving the number of loopless matroids of rank k on n labeled points (n >= 1, 1 <= k <= n). 7
1, 1, 1, 1, 4, 1, 1, 14, 11, 1, 1, 51, 106, 26, 1, 1, 202, 1232, 642, 57, 1, 1, 876, 22172, 28367, 3592, 120, 1, 1, 4139, 803583, 8274374, 991829, 19903, 247, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
From Petros Hadjicostas, Oct 09 2019: (Start)
The old references had some typos, some of which were corrected in the recent ones. Few additional typos were corrected here from the recent references. Here are some of the changes: T(5,2) = 31 --> 51; T(5,4) = 21 --> 26; sum of row n=5 is 185 (not 160 or 165); T(8,3) = 686515 --> 803583; T(8, 6) = 19904 --> 19903, and some others.
This triangular array is the same as A058710 except that it has no row n = 0 and no column k = 0.
(End)
LINKS
W. M. B. Dukes, Tables of matroids.
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
FORMULA
From Petros Hadjicostas, Oct 09 2019: (Start)
T(n,1) = 1 for n >= 1.
T(n,2) = Bell(n) - 1 = A000110(n) - 1 = A058692(n) for n >= 2.
T(n,3) = Sum_{i = 3..n} Stirling2(n,i) * (A056642(i) - 1) = Sum_{i = 3..n} A008277(n,i) * A058720(i,3) for n >= 3.
T(n,k) = Sum_{i = k..n} Stirling2(n,i) * A058720(i,k) for n >= k. [Dukes (2004), p. 3; see the equation with the Stirling numbers of the second kind.]
(End)
EXAMPLE
Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
1;
1, 1;
1, 4, 1;
1, 14, 11, 1;
1, 51, 106, 26, 1;
1, 202, 1232, 642, 57, 1;
1, 876, 22172, 28367, 3592, 120, 1;
1, 4139, 803583, 8274374, 991829, 19903, 247, 1;
...
CROSSREFS
Same as A058710 (except for row n=0 and column k=0).
Row sums give A058712.
Columns include (truncated versions of) A000012 (k=1), A058692 (k=2), A058715 (k=3).
Sequence in context: A267318 A050154 A179454 * A202906 A177984 A157013
KEYWORD
nonn,nice,tabl,more
AUTHOR
N. J. A. Sloane, Dec 31 2000
EXTENSIONS
Several values corrected by Petros Hadjicostas, Oct 09 2019
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)